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Unformatted text preview: Exam : 165 MATH 2451 thq_02 TC Use
Course : Multivariable Calculus Due Date : 20160126 Instructor : McCary KEZ (Flrst Name) (Last Name) 20160119 14 : 10 1. (10 points} Let. c, s E R be nonzero. (a) Find the real eigenvalues or Show there are none. Is ,4 (cc/mm MMLT up)
A ROTAT'MM. FM) =r/(Utr A 7!“ (JV/0 = (Acf‘ 4453’
P (Q) > 0 NO [(8, EMEWA Ls (b) Find the real eigenvalues or Show there are none. rm= pm=o =9 ¢1=1W r; A HyPE/zeoch 20THwa 1:5,) Mﬁw (A; mime» To an 2. (10 points} Evaluate. They all endup as “nice” expressions. You’ll see all of these later! (a) Easy. det [(20809) —T sin(9}] z A(m(9))x+jﬁ (9)): = ﬂ sin(9} r c0s(n‘9} (
(b) Medium. f EXPAND on) Mrs ooLuMA} c0s(n‘9} —r sin(n‘9} f} 44;.(9) J‘Lwﬂo) 095(9) vim“) (423(9) 7M0»
det sin(9} r c0s(n‘9} f} = 0 o  0' o D 1‘ /' may) j; WW)
I} f} 1 =1 (e) Hard: ﬁgure it out somewhere else and then neatly copy it here. .. sin(¢'} sin( } ,0 sin(¢'v} c0s(n‘9} ,0 c0s(¢'v} sin(n‘9} 5111(0) c0s(n‘9} —,0 5111(0) sin(9} ,0 c0s(¢'} c0s(n‘9} EXPAND 0"] TI”; '20“)
det n9
c0s(¢'} f} —,0 5111(0) ’ ,fMWMo) memre) I _ “meme? —,ow~(®M(9)
‘ “(PM [fwmmm ,mmmm ‘°'M['v]* (—rm(¢))0“"3 MWWKQ ,aMW)m(93 _M(¢)4§,.(e) NW) (NW) _ M(W‘M(°) 'MWlWW)
: (fault?) w) (M (a) MOP)M(0] '(“WW) MW)M‘M(0) W(\p)MH(97 =1)“th ( (wonmwmm + (WWW/mm») — WM (v) ((mmw on)“ + (w mﬂweﬂ“)
W L————'4
s  ,9“ msQO m mm) _ f,“ Mm (9mm?) =  ’* mm (MW + M<)(w( a“)
r ( have, = 1’me U 1 '
(a) Plot the parallelogram determined by Mg 51 and Mg 52 for the given value of t. . mm. . nan . inssa:. i=0 i=1 i=2 i=4 3. (10 points) Consider the matrix M3 = [1 1. known as a horizontal shear. It is very important to notice that det Mg : 1 which implies that M3 is areapreserving (all of your parallelograms
should have the same area, which is interesting because you horizontally shear the unit square out to t = 10'12
without changing its area). (b) Consider the matrix A = [6.5], where (i and I; are given in the plot. i. Determine the matrix R which will rigidly rotate the the indicated parallelogram such that the image of ti
will sit on the positive 33axis. That is, the columns of RA : [Rin] will determine the new, rotated
parallelogram. ii. Determine the matrix S which will horizontally shear RA into a rectangle. That is, columns of S RA will
determine a rectangle. G) & :sﬂ' 9v? go 131mm 8y ‘1}: = mi...‘ all J?»
g a, 42 E "a? 0‘1 a 7
Procedures such as this are related to matrix factorizations which are varied and important. The purpose here is to make sure you understand the associated geometry of what is happening and not just the (important, but
boring) steps of something like an L U factorization. 4. (10 points} The shaded regions are ellipses centered at the origin. {a} Find the matrix R which performs the following mapping, and compute det R.
an (4.2) vhf2) N we) we.» r;
a it _ E
a, (b) Find the matrix L which performs the following mapping, and compute detL. [a] {c} Using the previous parts, ﬁnd the matrix M which performs the following mapping, and computedetM. m 6 N5 0? 9112, r.
N E M 1‘5 or IN mm o @ M r:
R L
~—> w
[M] = 1.,[5 Elf °] W" 4 comm! EWW oA/ rm Wide:
M = WU ' 3 N Ms 1:? 51/
_ 4 [Era 4] 4 1
so  ’9' 3 9.13 M =L‘ E'
[M] = [R]"[L]" , .
(ﬁin MOW wHkT THIS WOULD MM!
= R '1 so . VTélM—LL/j
l] 09])Re(5:(a31_ 1 1
=[Rroa] New MO=H€ o
, 4 {2
(d) 'Nhat is the area of the tilted ellipse? ' L O ( mm CIPLLE)
DﬁrfﬂMINANT Is Me ewe mm. m 7% MM chum: = O (Dru/He) Am ® ‘77 Mlﬂllmﬁ’g = 9 THE “95”” ‘5 NOT 2* T’ﬁLTPDEmee, This page won’t. be graded. ...
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 Summer '09
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